Collisions between sodium atoms and nitrogen molecules

Volume 35, nllmbzr 3

COLLLSIOMS BETWEEN SODIM

CIIEhWAL

15 Ssptemter 1975

PHYSICS LETTERS

ATOMS AND NITROGEN MOLECULES

C. BOTTCHER Departmnt of llworctid

Physics. Sduster Laboratory, the University, !darrChcsterf.113 9pL, UK ad SCietlCt? Research Courrci!, Daresbury Laboratory, Daresbwy, Wat-rirg:on WA4 4AD, UK Received

13 June 1975

It is shown that the low energy scattering of electrons by nitrogen molecules may be described by 1 nonloul, energy dependent pseudopotential. This psecdopotentizl c;zn then be used to construct n model of the interaction beween XI r&k& atom in any state and a nitrogen mo!ecule in its gound electronic stnte. Potenti suzfzces are presented for the ground znd some excited states of the Na-N2 system. Cross sections are c2tculated for the elastic scattering of sodium atoms in the *S, *P states and for the spin-flip process 2P1,2-2P3,2, in each c3se with and without a chqe in the rota!ionai state of the nitrogen molecule. Isotope efkcts are briefly discussed. The thermal spin-flip cross section is in reasonable accord with expcriment.

1. Introduction The transfer

of energy from an electronicaily

ex-

cited species to a stable molecule is a process of great importance in many situations, lasers, atmospheric physics, photochemistry etc. [I] _This type of process is important because it is very efficient and we understand in a general way that this is connected with the large number of degrees of freedom, rotational and vibratiomal, possessed by even the simplest molecules. However the detziled mechanism of energy transfer is hardly understood at all. If the energy transferred is less than 1 eV or greater than 10 eV it is likely that an almost resonant energy difference exists in the molecular spectrum. But in the range from l-10 eV very few transitions with non-negligible Franck-Condon factors are available and we must invoke an electronic surface crossing mechanism: almost all molecules can formstable or unstable negative ions 121 so that if A(i,f) are two states of the atom A and M, hi- represent a molecule and its negative ion we can visualize the process A(i) + M -+ AfM-

+ A(f) + M

(I?

going through crossings of the surfaces AM, A%-.

Experimentally the most extensively studied system of this class is probably Na-$J2 [3,4]. Theoretical progress has been impeded by lack of knowledge of the NaN2 potential surfaces. In this paper we take the first step in a comprehensive study of Na-N, interactions by calculating potential surfaces with the R’, nuclei fixed in their equilibrium positions. These enable us to calculate elastic and spin-flip processes, possibly with rotational energy transfer. Vibrational energy transfer and quenching will be considered in later ~mmunications. Our caiculations are based on. an extension of the model potential method which has been extensively applied to atom-atom proceses [S-S]. Thus we can make use of the large body of experimental and theoretical data on e f N2 scattering [9,10] and in particular we take account of the famous shape

resonance at 2 eV.

2. Construction

of pseudopotentials

The model we use for NaM2 is that of tron outside

2

single elec-

the two closed shell systems Na+ and N2_

The interactior! of the electron with the Na* core is described, as before IS], by a local, energy indepen-

Any model of the process (1) must be capable of de-

dent, spherically

scribing the negative ion M-.

tron-N2

symmetric

interaction

model potentia!.

The elec-

cannot be spherically symmetric, 367

15 Septenber

CHEMICAL PHYSICS~LETTERS

Volume 35, number 3

1575

e.g.,we lmow that it contains long-range terms involving the quadrupole moment and anisotropic polarisability of Ni. A more serious problem is the representation of the shape resonance: it does not seem possible to fit a set of phase shifts, one of which goes through a multiple of r/2, by a single ener,Tdependent model potential

(the same difficulty

arises

in describing

the

Ramsauer effects in tile heavier inert gases)_ Thus we have introduced separate, energy-dependent, short-

range potentials for each partial wave. The partial waves are defined by expanding the wavefunction of the scattered

.electron

in spherical

harmonics,

the z-axis being along the line oinuclei. We shall consider only five terms Ln = so, po, du, pn, dn and neglect the couplings between them. The mode! harnil-

I

I

4 .B

!

.2

I

6

.4 k

tonian is (atomic unit:; will be used throughout) H=

-iv’

a0

--

a5

2r4

-_

(%Q +-%+ ?

2r4

Fig. 1. Vaiation of the range parameter B = aim for each of of wavenumber k. the potentials LJfil,,,as a function

p,(cose)

)

We took the short-range potentials to have the form (3)

u/m (r) = bl, (&,, =

where coo = 12, a2 = 4.2, Q = -1 .I [lo];

ws is a cut-

-

1) exp

[-@/Q,,,

)* 1 ,

0,

r < cl,r:

,

r > alrn

_

(7)

off function (4)

We adjusted the parameters aim and bin, to reproduce the diagonal elements of the K-matrix calculated by Burke and Chandra

[lo] ?. Eq. (6) was soLved to ob-

tain phase shifts nfrn such that whose argument ?I,,

is always x = 2 .O r in the present is the projection operator

case;

(5) If we substitute (2) into the Schrijdinger equation of

the hamiltonian (3) anintl neglect coupling between different I-states due to the long-range FI, satisfies

ian~~n,

= K&n,

1mj ,

@I

Fortunately the off-diagonal elements K(lm, inz) are fairly small. The phase shifts turned out to be much more sensitive to vartitions in al, than in blrri so we form = i. took blm = ti.0 for rrr = 0 and blrn = 4.0 The variation ofalln with wave number k is shown in fig. 1.

term we find that

3. Calculation of potential surfaces Let the Na nucleus

f The Kmazix

elmeats

See also fig. 5 of ref.

be

placed at A znd the centre

were taken from C3andra.s [lo].

Thesis.

Volume 3.5, number

of the NN line be placed at El, where the distance AB is equal to R and the angle between NN and AB is denoted by 0. The position vectors of the eiectron relative to A, B are P,, q., We make the approximation that the interaction is weakly dependent on 0 SO that A the projection of angular momentum on AB remains a good quantum number *_ This means that we can drop from the hamiltonian some terms involving eib. As in previous work the hamiltonian contains the interaction of the electron with each core, the core-core interaction and induced dipole terms. There is an extra induced dipole term involving the anisotropic polarisability. It is of course necessary to rotate the coordinates in (3) through an angle a. After some manipulation we find that the model hamiltonian is

+ I&&, where

15 September

CHEMICAL PHYSICS LETTERS

3

0) ,

Q,, 0) + &(R,

VA in the Naf model

(9)

potential,

1975

the hamiltonian 8) using a basis on A and B [5]. The non&a! operator 31,, is easily handled, as. in the work of Bardsley [13], but the energy dependence in ?i’i, requires a little more thou&t. Except at small distances the field of the,Naf ion is almost coulombic and the electronic eigenenergy in a given _molecular state is close to its atomic value, say -I Cl. Thus near the N2 core the kinetic energy of the electron is perWe diagonalised

set of Slater orbitals

fectly well-defied, E, = ICI - I/R

(13)

and this is the prescription we have used, At large values of R, EK becomes negative and we then used the zero-energy limit of Urn1 ; this should not lead to serious error since the overlap between the ?
+ ?V2(R)P2(cosO)

.

(14)

The coeffcient than 2 X 1O4, is obtained

oFP~ was found in all cases to be less so within the accuracy of the model it can be safely neglected. In table 1 we present the values of W,, and TV2 for the states 03s, 03~ and rr3p. I: is easily shown that the model correct!y predicts

from (3),

Vint(r, d, 0) = -+

c

w3cosL

-

y‘ 2

R’;!

+ $a4 (

contains

f-p5 ”

quadrupole

7

P~(COSO)

and induced

Vn,,(R,O) = V,,(R,O)

part of the interaction_

P,(ccsO)

There

are two

terms of order Rm6 , a van der Waals contribution and a quadrupole-quadrupole contribution, with coefficients given by

(11)

1

dipole

the long-range

w3 c0se

c vdW =CUo(qo+q2)

terms and

+

~~&of3~z)Pa(COS@)

CQQ = N&~ P@m)

1

,

(151

where qi = @PI)Na. At R = 30 the 03s and 53~ interactions are predominantly long-range while the 03~ interaction is still dominated by overlap forces.

-“o 2R4

+($ -~jP&osB).

(12)

Because

of the attractive

drr potential

electronic

chaqge

tends to cling to the N, cor’e. As in the alkali-inert

gas systems the thermal cross sections involving “P In (IO), dX&(@) is an element of the rotation matrix [ll], while in (12) VHF is the local Hartree-Fock interaction [ 121 which becomes important for R < 4. caimktions, putting in the neglected terms as a perturbation, szlows that ‘r states we not affected in the

atoms are 1argeIy determined by this overlap interaction in the w state. It is interesting that in alkali-&, Ne the interaction is repulsive while in alkaii--.4r, Kr, Xe and N2 it is attractive.

* More detded

first order wvhi!e II states ore. #it is the averaae value.

up. Whai we dcukte

he,-

4. Cross secttons

We are interested

in the processes 369’

CHEh!ICAL PHYSICS LETTERS

Vo!umk 35, number 3 Table 1 Na-N2 interaction

R

15 September 1975

energiCs r!J,(u3s)

IV*(03s)

Iv:!(ir3P)

ruo(a3Pl

r+‘,(a3p)

rvo(((r3p)

259E-1 1.93E-1 1.228-2

3.40E-2 3.3OE-2 1.73E-2

3.llE-1 2.12E-1 2.86E-:!

9.00E-3 3SOE-2 5.80E-3

3.19E-1 2.48E-2 5.245-2

4.00E-3 l.ODE-2 6.30E-3

8.44E-3

4.46E-3

2.28E-2

$.70E-3

-8.42E4

7.86E-3

3.8lE-3

2.80E-3

l.l6E-2

3.40E-3

-S .97E4

3_63E-3

8

1.75E-3 9.6lE4,

1.64E-3 1.1453

5.23E-3 1.92E-3

2.22E-3 1_6OE-3

-5.43E-4 -4.23E4

l.l4E-3 7.9154

10 12

650E-c

2.97E-4

-1.13E-4

l.l9E-3

-1.43E-G

8.0455

4.40E-5

l.BOE-4

-3.4lE-4

4.73E-4

-7.02E-5

3.54E-5

14

4.82E4

3.66E-5

-2.26E4

1.48E34

-3.628-5

1.78E-5

3.18E-5

-1_90E-S

8.15E6

2 3 4 5 6 7

-7.12Ed

-5.56Ed

-1.23Ea

:“o

-2.82

25 30

Ed -8.9OE-5

-1 SOE6 --3.60E-7

-3.34E-5 -7.3486

-4SOE36 -2.12E6

-6.05Ed -1 J36E-6

1.54E-6 2.4OE-7

-3.30E-7

-1.30E-7

-2.15E36

-7.30s7

-6.60E-7

SSOE-8

2PJ)+ N2(j+K)

+ Na(2S, 2PJ.) , (16)

N2(j)+ Na(2S,

where I\_’= a,+?. The approach we use is simiiar to that of iefS. 18,141 in that a11 quantities are expressed in terms of elastic phase. shifts. The separation between rotational leveis in N2 i; so much smaller than thermal energies that we are justified in using the impu!se approximatior. for the rotational part of the transition. Thus the cross section for “elastic” scattering by an orientation dependent potential V(R) is

We have expanded 1 Q,

=z

v = LTo+ V,P2

and written

+s

v,(R)P[(cosO)dz

.

To extend the= formlllae into the strong coupling regime requires more thought. The sum of ‘YK must not exceed unity and whenever all the I& I are large, all the pK become equal. Thus we define 3)~ = sin2& sin 2[(c

&>,‘*I

[csii&JL1*,(X1)

We have applied (18)-(20) to the surfaces X, B 2Z and A 211 in table 1, to obtain oK(X, B, A) and also

where 277(b) = u-l jT_dzV(b +&), b is the usual impact parameter and & is a unit vector perpendicular to b. As a final simplification we are typically concerned with values ofj=

10 so thati%>.

First we examine the weak coupling situation, where all the interactions are small. After some algebra we find that the cross action in (17) depends only CI~ K and can be written

OK(B-A) involving the difference between VB and VA. In fig . 2 we plot the quantities of greatest physical interest, viz., the elastic ground state cross sections GK(%) = oK(X), the mean excited state cross sections Us = EGG f $G=(A) and the spin-flip cross section OK(SF) = OK(~P,,~ -2P3,t) = &OK@-A). If rotationally inelastic transitions are not distinguished one observes a tots1 cross section u = u. t 2g2. Thus

the pressure broadening of the Na(D) line in an airno., sphere of N, can be calculated from u(2P). It can be seen that the cross sections have the usual oscillatory structure, more pronounced in the rota!ionally inektic transitions. For the ground state process u2 is comparabIe with Go, but for the excited state p:ocesses

oO is m&h

larger; this behaviour

is an

immediate consequence of the large overlap force in

Volume 35, number 3

CHEMICAL

PHYSICS LEiTERS

15 September 1975

AcknswIedgement

1 have pleasure in thanking Professor P.G. Ehrke and Dr. B-D. Buckley for discussions on the e-N2 scattering model.

[l] H.S.W.

Massey and E.H.S. Barhop, Electronic and ionic impact phenomena; Vol. 3 (Oxford Univ. Press, iondon,

I

1973).

.

Id

Id T(deg1

(21 C.D. Cooper and R.N. Compfon, J. Chem. Whys. 59 (1973) 3.55@. [3] hl. Stupavsky and L. Krause, Can. J. Phys, 46 (1968)

IO'

Fig. 2. Dependence of Na-N2 cross sections on temperature: S, ground state elastic; P, excited state elastic; SF, sp&flip l/2 -3/Z; K = 0, rotationally elastic; K = 2, rorationally inelastic.

the B ?Z state. At 400 IS we predict o(SF) =405 17; compared with the experimental value 5 14 02 [3]. This is reasonable agreement in view of the uncertain-

ties in both the scattering calcclation and the mezsureIsotopic effects are easily studied experimentally and have aroused considerable interest_ In order to

ments.

gauge their possible

significance

we examined

collisions

in which 14N15N collided with Na. Transitions with Aj = 1 are now possible and the cross sections u1 (S), aI (P) and crl (SF) are surprisingly large, viz., 107, 172 and 105 0: respectively. It is hoped to continue this work by looking at other atom-molecule pairs, at quenching and vibrational energy transfer and by carrying out fully quanta1 scattering calculations.

2127. 141 P.L. Lijusc and R.J. Eknaar, J. Quant. Spectry. Radiative Transfer 12 (1972) 11!5. [S] C. Bottcher, A. Dalgarno and E.L. Wright, Phyr Rev. A7 (1973) 1606. [6] C. Bottcher, Chem. Phys. Letters 18 (1973) 457. [7] C. Bottcher aiid A. DaIgarno, Rot. Roy. Sot. A340 (1974) 187. [8] C. Bottcher, T.E. Cravens and A. Dalgtrno, Proc. Roy. Sot. A (1975), to 5e published. [9] G.J. Schultz, Rev. Mod. Phys. 45 (1973) 423. [lo] P-G. Burke and N. Chandra, J. Phys. 95 (lz172) 1696. [ 111 A.R. Edmonds, Angular momentum in quantum mechanics (princeton Univ. Press, Princeton, 1960). [!2j F.H.M. F&al, J. Phys. B3 (1970) 635. [ 13 1 J.N. Bardsley, in: &SC studies in atomic physics, Vol. 4, eds. E.W. McDaniel and MM!. McDowell (NorthHolland, Amsterdam, 1974) p. 299. [14] S. Wofsy, R.H.G. Reid and A. Da&no, Astrophys. J. 168 (1971) 161.

371